Coarse-graining two-dimensional turbulence via dynamical optimization

TL;DR

This paper introduces a new model reduction method for 2D turbulence using an optimization principle, resulting in a coarse-grained model that accurately predicts dynamics without adjustable parameters.

Contribution

It develops a novel optimization-based closure technique for 2D turbulence that captures key dynamics and is validated against numerical simulations.

Findings

The method accurately predicts the evolution of low modes.

It includes a scale-dependent eddy viscosity.

No adjustable parameters are needed for validation.

Abstract

A model reduction technique based on an optimization principle is employed to coarse-grain inviscid, incompressible fluid dynamics in two dimensions. In this reduction the spectrally-truncated vorticity equation defines the microdynamics, while the macroscopic state space consists of quasi-equilibrium trial probability densities on the microscopic phase space, which are parameterized by the means and variances of the low modes of the vorticity. A macroscopic path therefore represents a coarse-grained approximation to the evolution of a nonequilibrium ensemble of microscopic solutions. Closure in terms of the vector of resolved variables, namely, the means and variances of the low modes, is achieved by minimizing over all feasible paths the time integral of their mean-squared residual with respect to the Liouville equation. The equations governing the optimal path are deduced from Hamilton-Jacobi theory. The coarse-grained dynamics derived by this optimization technique contains a scale-dependent eddy viscosity, modified nonlinear interactions between the low mode means, and a nonlinear coupling between the mean and variance of each low mode. The predictive skill of this optimal closure is validated quantitatively by comparing it against direct numerical simulations. These tests show that good agreement is achieved without adjusting any closure parameters.